The proof of the affine \(L_p\)  Pólya-Szegö principle in [1]} mainly relies on the affine \(L_p\)  Petty projection inequality from [2]} and the solution of the normalized \(L_p\)  Minkowski problem [3]}. Haberl, Schuster and Xiao in [4]} mainly relies on the Haberl-Schuster version of the affine isoperimetric inequality [5]} and the solution of the normalized \(L_p\)  Minkowski problem [3]} to prove the asymmetric affine Pólya-Szegö principle. The discovery of the Haberl-Schuster version of the \(L_p\)  Petty projection inequality was made possible through the important advances in valuation theory by Ludwig [7]}, [8]}. Inspired by the work of Haddad, Jiménez, and Montenegro [9]}, Nguyen [10]} made use of the general \(L_p\)  Busemann-Petty centroid inequality [2]} to prove the general affine Pólya-Szegö-type principle.
